Rate-synchrony relationship between input and output of spike trains in neuronal networks

Neuronal networks interact via spike trains. How the spike trains are transformed by neuronal networks is critical for understanding the underlying mechanism of information processing in the nervous system. Both the rate and synchrony of the spikes can affect the transmission, while the relationship between them has not been fully understood. Here we investigate the mapping between input and output spike trains of a neuronal network in terms of firing rate and synchrony. With large enough input rate, the working mode of the neurons is gradually changed from temporal integrators into coincidence detectors when the synchrony degree of input spike trains increases. Since the membrane potentials of the neurons can be depolarized to near the firing threshold by uncorrelated input spikes, small input synchrony can cause great output synchrony. On the other hand, the synchrony in the output may be reduced when the input rate is too small. The case of the feedforward network can be regarded as iterative process of such an input-output relationship. The activity in deep layers of the feedforward network is in an all-or-none manner depending on the input rate and synchrony.

Figure 1

(a) Schematic illustrating how the input spike trains are transformed by neuronal networks. (b) The input synchrony $S_{in}$ versus the correlation coefficient $c$ with $R_{in}=22$. (c) The input synchrony $S_{in}$ versus the input rate $R_{in}$ with $c=0.1$.

Figure 2

$R_{in}=31$, $c=0.1$, $S_{in}=0.063$, and $g=60$. (a) The input STH $p_{in}\left(t\right)$ versus time. (b) Spike raster plots of the output spike trains. (c) The output STH $p_{out}\left(t\right)$ versus time. (d) The membrane potential $V_i$ of one selected output neuron versus time. (e) The coefficient of variation $C_v$ of each neuron versus its firing rate. (f) The power spectrum of $p_{out}\left(t\right)$.

Figure 3

$g=60$. (a) The output rate $R_{out}$. (b) The output synchrony $S_{out}$. (c) The coefficient of variation $C_v$ plotted as a function of the input synchrony $S_{in}$. Each symbol represents different input rate $R_{in}$: 5 (○), 13 (▽), 22 (×), and 31 (◇). The dotted line in (b) is the diagonal $S_{in}=S_{out}$. Inset of (c) shows $C_v$ versus $R_{in}$ with $S_{in}=1$.

Figure 4

$R_{in}=13$. (a) The output rate $R_{out}$. (b) The output synchrony $S_{out}$. (c) The coefficient of variation $C_v$ plotted as a function of the input synchrony $S_{in}$. Each symbol represents different coupling strength $g$: 24 (○), 30 (▽), 45 (×), and 60 (◇). The dotted line in (b) is the diagonal $S_{in}=S_{out}$.

Figure 5

(a) The output rate $R_{out}$. (b) The output synchrony $S_{out}$. (c) The coefficient of variation $C_v$ plotted as a function of the coupling strength $g$. Each symbol represents different input parameters: $R_{in}=13$, $c=0.1$, and $S_{in}=0.043$ (○); $R_{in}=22$, $c=0.1$, and $S_{in}=0.057$ (▽); $R_{in}=31$, $c=0.1$, and $S_{in}=0.063$ (×); and $R_{in}=22$, $c=0.6$, and $S_{in}=0.372$ (◇).

Figure 6

(a) Each curve represents different parameters: $R_{in}=31$, $c=0$, $S_{in}=0$, and $g=45$ (○); $R_{in}=22$, $c=0.1$, $S_{in}=0.057$, and $g=45$ (▽); $R_{in}=13$, $c=0.3$, $S_{in}=0.134$, and $g=30$ (×); and $R_{in}=5$, $c=0.5$, $S_{in}=0.199$, and $g=30$ (◇). The layer index increases along the direction of the arrows. (b) The synchrony $S_{10}$ versus the input synchrony $S_{in}$ with $g=30$. Each symbol represents different input rate $R_{in}$: 5 (○), 13 (▽), 22 (×), and 31 (◇).